https://file.aiquickdraw.com/mathbot/file/1727188182795owe1e8du.jpg

In the image provided, the angles \( (3x - 7)^\circ \) and \( (x + 5)^\circ \) are given as alternate interior angles because they are formed by two parallel lines and a transversal. Alternate interior angles are equal, which gives us the equation:

\[
3x - 7 = x + 5
\]

### Step-by-step solution:

1. **Solve for \( x \)**:
\[
3x - 7 = x + 5
\]
Subtract \( x \) from both sides:
\[
2x - 7 = 5
\]
Add 7 to both sides:
\[
2x = 12
\]
Divide both sides by 2:
\[
x = 6
\]

2. **Find \( y \)**:
Substitute \( x = 6 \) back into the expression for \( y = 3x - 7 \):
\[
y = 3(6) - 7 = 18 - 7 = 11
\]

Thus, \( x = 6 \) and \( y = 11 \).

Would you like more details on any step, or have any further questions?

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Here are five related questions you could explore next:
1. What are alternate exterior angles and how do they differ from alternate interior angles?
2. Can you solve a similar problem with supplementary angles instead of equal angles?
3. What happens to the angles if the lines are not parallel?
4. How would you solve for \( x \) if you had consecutive interior angles instead?
5. Can you find the values of \( x \) and \( y \) if the angles were complementary?

**Tip:** Remember that alternate interior angles are equal only when the lines are parallel!

Find x and y (If you get a decimal it is ok!) given that (3x - 7)° and (x + 5)° are alternate interior angles.

This problem involves solving for x and y using alternate interior angles formed by two parallel lines and a transversal. The angles (3x - 7)° and (x + 5)° are equal. Step-by-step algebraic manipulation reveals that x = 6 and y = 11. Suitable for Grades 7-9.

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